Study of the chiral transition in SU(2) lattice gauge-Higgs theory with dynamical fermions

30 June 1988

PHYSICS LETTERS B

Volume 207, number 4

STUDY OF THE CHIRAL TRANSITION WITH DYNAMICAL FERMIONS

IN SU(2) LATTICE GAUGE-HIGGS

THEORY

Smya AOKI, I-Hslu LEE Physm Department

Brookharen

National Laboratorv.

C’pton NY 11973, C’S4

and Robert E SHROCK Instmtie for Theoretml

Phvsm, State Unwmtty

of New York, Stonv Brook, NY I1 794, USA

Received 14 Aprd 1988

We report the first measurements, with fully dynamlcal fermlons, of the (zero-temperature) choral symmetry-breakmg phase transition ln the SU (2) part ofthe standard SU (2) x U ( I) electroweak lattice gauge theory These are performed with a Langevm algorithm Good agreement ISfound with the previous analytic strong gauge coupling results regarding both the location and the order of the transition The choral boundary 1s also measured m the interior of the phase diagram and the phase structure with dynamical fermlons ISdiscussed

quent numerical studies [6,8] have also used quenched fermlons The existence, location, and order of the transition have been established analytltally [ 6 ] m the llmlt of strong gauge couplmg, &= 0, using a type of mean field method which has also been applied quite successfully to the U ( 1) gauge-Hlggs fermion theory [2,3,5] This analytic method goes beyond the quenched approxlmahon and includes some effects of dynamical fermlons, as was proved [ 3 ] by expllcltly calculating the average gauge-Hlggs interaction (L)=(Re(~t,U,.~,+,,,)) at&=Oand showing that it was non-analytic at the choral transltlon (it is analytic for quenched fermlons) As was pointed out m ref [ 31, since the small-p, expansion has (at least) a finite radius of convergence, the transition at @,=O 1s the endpoint of a lme of transitions which exist for &> 0, and this lme cannot end anywhere m the mtenor#‘, hence, the choral boundary

In this letter we report the first study of the spontaneous choral symmetry-breakmg (SxSB) phase transition m the SU (2) lattice gauge-Hlggs theory with I= l/2 Hlggs fields and I= l/2 dynamical fermlons The importance of including fully dynamical fermlons 1s clear, as had been discussed m our earller work [l-7] The SU(2) theory with I= l/2 Hlggs and fermlon fields 1s of direct physlcal significance, smce it 1s the SU(2) sector of the standard SU( 2) x U( 1) electroweak theory The phase structure of the lattice theory plays a fundamental role m determining the fermlomc mass spectrum only m the phase where choral symmetry 1s not spontaneously broken can the physical fermlons be naturally light compared to the electroweak scale of- lo* GeV Hence, a continuum limit of the lattice theory which yields properties m agreement with experiment should be taken from wlthm the phase without spontaneous choral symmetry breaking Indeed, this may provide an ab mltlo answer to the longstandmg question of why the SU (2)-nonsmglet fermlons m nature have weak lsospm Z= l/2 [ 41 The choral transltlon in this theory was first observed m numerlcal slmulations using quenched fermlons [ 11, and subse0370-2693/88/s ( North-Holland

03 50 0 Elsevler Science Publishers Physics Pubhshmg Dlvlslon )

PI This IStrue smce II the hne of choral phase transitIons did end m the interior, then one could analytlcally continue the function representmg (a) from the chorally symmetric phase, where It vanishes Identically, to the chorally broken phase This would contradict the fact thdt it 1s nonvamshrng In the latter phase

BV

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completely separates the phase diagram into two phases, characterized, respectively by explicit reahzatlon of the choral symmetry and by the spontaneous breaking of this symmetry [l-3,6] Furthermore, the lattice approach provides a powerful nonperturbatlve method for studying models of weak mteractlons m which the SU(2) group IS strongly coupled [ 9, lo] and, as we chscussed before [ 2,4,7], our results suggested possible problems with the assumptions underlymg such models Another lmphcation IS that the choral transition obviously changes the renormalization group flow from then form m the bosomc gauge-Hlggs theory and affects the contmuum limit of the lattice theory [4,6,7] Given these far-reaching effects of fermlons on the SU( 2) gaugeHlggs theory, it IS obv;ously important to carry out a study with fully dynamical fermlon fields We recall that the theory ISdefined, m standard notation, by the path integral

s

z= n dun,

d#,, d#indxndL e-‘,

(Ia)

n(t

Udf &=4/g’ > 0, P= 4 Trf Upiaq), and, m our notatIona convention#*, y= 2ph The lattice IS formulated on a ~-dimensIona euchdean hypercubic lattice with d=4 The lattice spacing a will generally be set to umty The Hlggs fields Q,satisfy @in@, = 1, as IS well known, this does not imply that the contmuum Hlggs fields are of fixed length With no loss of generality, we take j&> 0 We use staggered fermlons [ 111 xn because of their advantages for studies of choral symmetry ti3 The qns={l forp=l, (-I)“‘+ +nJ+’for2
” The alternate conventIon ys/$, IS used m a number of papers on gauge-Hlggs models tn As IS well known, the staggered fertmon formulation yields 2*/2’d”1 species of Dlrac fernuons, I e ,4 for d=4

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30 June 1988

manner to the gauge fields, this does not reduce the physical relevance of the model, smce the SU (2) sector of the standard electroweak theory can be rewntten in a vector-like manner [ 13 ] (essentially because the representations of SU (2) are real) The theory ( 1) IS simulated by means of a Langevm algorithm [ 14,151 for gauge, Hlggs, and fermion fields We work with a 6j lattice with standard boundary condltlons periodic for bosom and antiperiodic for fermlons The value Ar=O 01 was used for the Langevm time step, as m Langevm slmulatlons of QCD [ IS ] For each value of fig and ph, we used both ordered and disordered mltlal configur&Ions to check convergence The conjugate gradient method ISused to calculate the inverse ofthe fermlon operator Q which enters m the actlon as S,= Z,J,Q,x, Measurements of (fi),a3= (.X,Z_,,& x,yJ ),,,a3~ 2M,,, were performed for ma=0 1 and ma = 0 05 From these an extrapolation to m = 0 was carried out to obtain the spontaneous choral condensate (~)a”=llm,,_o+ (%),,a3 =22Mg4 The runs involved a number NEof Langevm lattice updates for mltlal equilibration followed by a number NM of updates with concomitant measurements, these varied from NE= 500, NM= 2000 to NE= 9000, NM= 12 000, where the lengths depended on the values of & and j?,, (and were greater m crltrcal regions) As is well known [ 15 ] , a Langevm simulation with finite ATactually yields a statlstrcal ensemble correspondmg to an action which differs shghtly, by terms of order AZ, from the mput actlon With the value Az=O 01, this ISnot expected to be a large effect m our data In fig la we present our measurements of (~)~a~ for&=0 and ma=0 1 and ma=0 05 The extrapolation for the spontaneous choral condensate 1s shown m fig lb This data gives strong evidence for a contmuous (second-order) SXSB phase transitlon m the thermodynamic hmlt, on the 64 lattice, we obtam the crrttcal couphng /.$,,(&=0)=2 520 3 (There would, of course, be a small fimte-size shift m this quantity as the lattice volume Increases to mfimty ) This value IS close to the crItica value found by analytic methods m ref [ 6 1, VIZ, I%,c(& = 0 )MF = 2 76 rt:0 02 This IS roughly the level of agreement that one would expect from experience m statistical

44Recall

that the sign of {a)

IS a conventIon

PHYSICS

Volume 207, number 4

LETTERS

30 June 1988

B 08

b

a 07

06

06

i

05

r:

04

03

t 02

t

0

01

. I”‘(“‘*

40

t 00

i,t,,,

-r ,~lll.l~,,,l.,l,.,,,,,,O

0

0

/

10

20

30

,,,,,,:,,,,,,,,*

40

0

&

Fig

1 MS,,=$(;7X),,a3for

(a) ma=0

1 (o),OO5(*),

mechamcP The agreement 1s also expected m view of the fact that the analytic results of ref 161, hke those [ 2,3,5] for U( 1 ), did Include some dynamical fermlon effects For a given ,&, where (a) IS nonzero, it is slightly smaller than the correspondmg value obtained (again at /3, = 0) m the quenched slmulatlons [6] Our value of &,C(&=O) with dynamtcal fermlons 1s slightly smaller than, but close to the

USTo be a httle more quantttattve, recall that the spontaneous choral symmetry breakmg here essentially mvolves an axial LJ( 1) group As a guide, one thus might consider the 4D LJ ( 1) = 0 (2) spin model, for this model (on a hypercublc lattlce), mean field theory predicts a crmcal pomt PCMFT which 15 about 15% below the actual PC Thus, m both the spm and gauge cases, mean field theory overestimates the size of the reglen where the s~rnrn~t~ IS spontaneously broken by stmtlar amounts (As noted tn ref [2], however, this breakmg takes place for/L & ) (Note added When this work was essentially complete, we received a preprint by Kogut and Dagotto [ 171 m which a slmulatlon of U( 1) gauge-Hlggs theory with dynamlcal fermlons IS reported, these authors also find excellent agreement with our anaIytlc strong-coupling results of ref [ 2 ] We thank E Dagotto for sending us this preprmt )

(b) spontaneouscondensate.~~f(~)~~(~),allatli,=O

quenched value [6] phC (&=O)quen =2 75*0 2 The analytic study, quenched simulation, and present dy namlcal slmulatlon are also all m agreement as to the second-order nature of the choral transItion here In fig 2 we show measurements of (L) and (P) for p,= 0 and ma=0 1 and 0 0.5 The data for these two values of ma (and hence also the extrapolated values for m = 0) are essentially identical, showing that for &=O these quantmes are very msensrtlve to tn for small m In the absence of fermions, (L),,,(&=O)=r and (P)bos(j3g=O)=r4, where r=l*(y)/J,(y) and Z”(y) IS the modified Bessel function These functions are included in fig 2 for comparison, and one can see that dynamical fermlons do not cause a large change in either (ii) or (P) relative to their values m the pure bosomc theory Since for m =O, the (reduced) free energy at the choral f= l~mVOl_ouN; ’ In Z 1s non-analytic transition, it follows that its derivatives, m partlcular, (L) = ( 1/d)af lay, are also non-analytic there This non-analytlclty 1s evldentiy a small effect m the data Agam, this IS not surpnsmg, smce the mean field methods yielded a cusp smgularlty for (L), and the 473

Volume 207, number

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LETTERS

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30 June 1988

LO

09

08

07

06

05

04

03

02

01 m _-’

00

m-

0

./,,,,,

--_.,.......I...-

10

20

30

40

00

FIN 2 (L) for ma=0 1 (o), 0 05 (o), (P) for ma=0 0 05 ( n ), all at p,= 0 Sohd and dashed curves represent and (0 hos. respectively, at /I,= 0

I (0) (L > bol

explicit calculation for the U ( 1) case indicated that the amplitude of the cusp was rather small [ 3 ] Next, we proceed to j?, > 0 Our measurements of (&),,,a3 at &=O 5 and ma=0 1, 0 05 are quahtatlvely similar to those at &=O and hence are not shown here The values of (a) obtained by extrapolation to m = 0 from these measurements are shown m fig 3 As was the case at &=O, (xx) 1s slightly smaller with dynamical fermtons than m the quenched data [ 1 ] The choral transition 1s again continuous and takes place at p,, C(&= 0 5 ) = 1 65 It: 0 2 This critical value 1s slightly smaller than the quenched value&, ?(&=O 5),,,,=2 O&O 2 The behavior of (L) and (P) 1s similar to that at &=O and is not shown The fact that (fix), ( L) , and (P) all behave qualitatively m a manner similar to that at &=O 1s implied by our previous analytic argument regarding the small-& expansion In order to investigate the confinement--Hlggs transition m the presence of dynamical fermlons and, m particular, whether it comcldes with the choral 474

05

10

15

20

25

0

Bh

Bh

FIN 3 M=f(X/oa’at/&.=O

5

boundary, we have also made measurements at /3,= 1 9, a value where, in the pure bosomc gaugeHlggs theory this confinement-Hlggs transition 1s present (at Ph=O 54) The measurements of (a) (see fig 4) and the resultant extrapolation to m=O, indicate that the chual transition occurs at phc(&=l 9)=040&005 Thedataon (L) (fig 5) and (P) show evidence for a phase transitlon at the same value of b,,, to wlthm the uncertainties of the measurements We do not observe any other transltlon, as a function of/J at this value of/?, Thus, these measurements suggested, to within their accuracy, that the confinement-Hlggs and choral transitIons comclde here, as they did m the quenched measurements at the nearby value&.=2 3 [ 1,8] For comparison, m fig 5 we also plot the values of (L),,,, at j&=0 (which we found to be very close to (L) with dynamical fermlons) and the full (L) at &=CC which 1s the same as m the bosomc theory [ 161 One can see that (L) has very similar behavior for small /3h at different DE (L) (&=O) = (L) (~~=O)bos=y/4-y3/96+O(y’), which is very close to (L) (~~,-co)=y/4+y’/12+O(y5) More-

Volume 207, number

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30 June 1988

R 10

040

09 035 06 030 07 025

E

06 A r-1 05 V

020

x

04 015 03 010 02 005

01

ooc 2

I,,03

00 04

05

06

3

07

01

02

Fig 4 izf,,zz+ (a),,,u’at&=l

9forma=O

03

04

05

06

07

a

Btl

Btl

1 (0),005

(0)

Fig 5 (t) CL),,,

over, one expects that at fixed j3,,, (L) IS an mcreasmg function of /3,, and these series expansions are m agreement with this inequality It is therefore expected that at intermediate values of &, (L) =y/4+O(y3) for smallp,,, and this 1s borne out by our data As IS evident from fig 4, the choral transition 1s more abrupt at &= 1 9 than at either &=O or 0 5 Indeed, we observe a two-state signal (ordered and disordered starts falling to come together) which lasted for about 9000 Langevm time steps m (a) (L), and (P) at this transition In addition, we see some mdlcatlons of tunneling occurring between the two states in these respective quantities This suggests that at &= 1 9 the choral transition might be weakly first order If this 1s the case, then there 1s a trlcritlcal point or critical endpoint on the choral boundary between &=O 5 and &= 1 9 separating second-order and first-order segments of the choral boundary Our measurements are consistent with the following general inequalities I (fi) 1, where it 1s nonzero, 1s a monotonically decreasing function of 1j?,,1at fixed

at&=1

t&=0)

9forma=O

1 (0),005

(0)

Solldcurvels

and cl data 1s CL) U&=m)

and a monotonically decreasing function of & at fixed j$, This behavior 1s m accord with the fundamental physical origin of (zero-temperature) spontaneous choral symmetry breaking m gauge-Hlggs theories this is due to the gauge-fermion mteraction Now, as noted before [ 2-7 1, as /i’,,increases at fixed &, the gauge-Hlggs couplmg increasingly restricts the gauge-field configurations which make a significant contrlbutlon to the path integral, indeed, as &co, the gauge degrees of freedom are frozen out and the theory reduces to that of a free fermion For sufficiently large Ph the gauge-fermion interaction is no longer strong enough to produce QSB, and the choral symmetry 1s restored A similar restriction of the slgmficant gauge-field configurations occurs as j$ mcreases at fixed&, due to the S, term The inequalities stated above are thus easily understood One should, however, recall a basic difference m the behavior of (xx) as a function of /3,, and &, m contrast to the choral transition which occurs, e g , at p, = 0 as a function of Ph, present theoretical and slmulatlonal results indicate that m the usual SU (2) gauge-fermlon system at j& = 0, (B) a 3 is nonzero for all finite j3, &

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(Renormalization group arguments indicate that m the asymptotic scaling regime asj+co, this quantity vanishes like < -3 -pi exp ( -c& ), where {IS the correlation length and p and c are certain positive constants ) Now (for arbitrary fermlon representation), at &=co, since the gauge degrees of freedom are frozen out, the theory reduces to a direct product given by Z(P, =m, Y) =Zo(4)

(Y)-Gf

>

(2)

where ZoC4) (y) is the partmon function for an O(4) spin model with spm-spin coupling y, and Z, is the partltlon function for a free fermlon Hence, m this hmlt, the fermlons decouple Several important corollaries follow from this First, (L) (&=co, y) = CL) (Y)bos, 1 e , at &=CXI, m the full theory with dynamical fermlons, (L) ISthe same function ofy= 2/3,, as it IS m the bosomc gauge-Hlggs theory (where It 1s given by the nearest-neighbor spin-spin correlation function of the O(4) model) Now at pg=co, the 0 (4) spm model has a (second-order) phase transltlon mvolvmg spontaneous breaking of the global 0 (4) symmetry In the gauge-Hlggs theory without fermlons, or with quenched fermlons, this transition is the endpoint of a hne of transitions extending into the interior of the phase diagram, namely, the confinement-Hlggs phase boundary (which ends m the interior) Moreover, since massless free fermlons do not spontaneously break choral symmetry, (a) = 0 V /?,, at & = co, so that there 1s no spontaneous choral symmetry breaking transition at any finite /I,, fol &= co However, the &+cc limit is smgular, as can be seen, e g , by the fact that the symmetry of the theory changes (the local SU (2 ) gauge invariance is replaced by a global O(4) symmetry) This singular limit 1s manifested, for example, m the fact that but ($)#O at &=co for (@>=OvP,<~ Ph>PhC0(4) Another analytic result IS that for small &,, one can integrate out the Hlggs fields and obtain a pure gauge-fermlon theory with a slightly shifted effective coupling (&)eff=/39+2-6pfi One thus expect that at nonzero but sufficiently small p,,,(B) again vanishes exponentially fast as &-+co Our measurements of the choral transition with dynamical fermlons are summarized m the phase dlagram shown in fig 6 This diagram 1s in agreement with the one inferred from our earlier analytic and quenched studies [ 1,4,6,7 ] However, it is on a firmer 476

00

10

20

05

6, Fig 6 Phase diagram of 4D SU(2) gauge-Hlggs theory with I= I /2 Hlggs and dynamlcal fermlon fields summarlzmg current measurements Choral boundary completely separates the phase diagram mto two phases Pomt at &=cc IS the crItIca pomt of the O(4) model

footing, since It 1s derived from a slmulatlon of the full theory and 1s not restricted to the quenched approxlmatlon This work 1s of direct importance for our understanding of (the SU( 2) part of the) standard electroweak theory First, it maps out the boundary of the phase in which choral symmetry 1s not spontaneously broken, which 1s the phase m which one should approach the continuum limit Second, since our dynamical fermlon slmulatlons yield the same phase structure as our earlier studies, they confirm the problems which we noted previously [ 2,4,7] with the strongly coupled standard model [ 9,101 This model assumes that the SU(2) (a) 1s strongly coupled, with a resultant spectrum of bound states, but (b) at the same time, choral symmetry is not spontaneously broken (since otherwise the fermlons would have huge dynamically generated masses of order the electroweak scale of - 1O2GeV and electric charge would not be conserved) We found that these two assumptions do not appear to be simultaneously satisfied the Hlggs-like phase where the choral symmetry 1s not spontaneously broken 1s completely separated by a nonanalytic boundary from the phase connected to the confinement phase of the pure gauge theory and does not appear to exhibit the spectrum of bound states expected for a strongly coupled theory (as was shown m ref [ 1 ] and recently confirmed m ref [ 81) The small-P, phase, which does exhlblt such a spectrum of strongly coupled bound states, also

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spontaneously breaks choral symmetry A third lmphcatlon of the present work 1s that it lends support to the possible explanation made earlier based on analytic and quenched studies for why the SU( 2) nonsinglet fermlons have weak I= l/2 [ 41 Finally, since at least part of the chu-al boundary 1s second-order, it would be of interest to examine it from the point of view of a candidate for a continuum limit [ 41 away from @,= co Additional results, including data for I= 1 fermlons, will be reported elsewhere We thank S Samelevlcl for dlscusslons of Langevm slmulatlons This research was supported m part by the US Department of Energy and the National Science Foundation under contracts DE-ACOZ76CH00016 (S A and I H L ) and NSF-PHY-85-07627 (R S) The computations were carried out on the Florida State Umverslty Cyber 205 and the Pittsburgh Supercomputer Center CRAY X-MP/48 via respective time allocations from DOE (S A , IHL)andfromNSF(RS)

References [ I] I-H Lee and J Shlgemltsu, Phys Lett B 178 ( 1986) 93 [2] I-H Lee and R E Shrock, Phys Rev Lett 59 (1987) 14

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[ 31 I-H Lee and R E Shrock, Nucl Phys B 290 ( 1987) 275 [4] I-H Lee and R E Shrock, Phys Lett B 196 (1987) 82 [ 51I-H Lee and R E Shrock, Phys Lett B 199 (1987) 541 [6] I-H Lee and R E Shrock, Phys Lett B 201 (1988) 497 [ 71 I-H Lee, in Proc Intern Symp on Field theory on the lattlce (Sedlac, France, 1987), Nucl Phys B (Proc Suppl ) 4 (1988) 438, R E Shrock, m Proc Intern Symp on Field theory on the lattice (Selllac, France, 1987), Nucl Phys B (Proc Suppl ) 4(1988)373 [ 81 A De and J Shlgemltsu, OSU preprmt DOE/ER/Ol545398 (1987) [9] L Abbott and E Farhl, Phys Lett B 101 (1981) 69, Nucl Phys B 189 (1981) 547 [ lo] M Claudson, E Farhl and R L Jaffe, Phys Rev D 34 (1986) 873 [ 111 T Banks, L Susskmd and J Kogut, Phys Rev D 13 ( 1977) 1043, L Susskmd, Phys Rev D 16 (1977) 3031, J Kogut, Rev Mod Phys 55 (1983) 775 [12]N KawamotoandJ Smit,Nucl Phys B192 (1981) 100 [13]HM Georgl,ascltedmref [lo] [14]G ParlslandYS Wu,Scl Sm 24(1981)483, 1 Drummond, S Duane and R P Horgan, Nucl Phys B 220(1983)119 [ 151 G Batroum, G Katz, A Kronfeld, G P Lepage, B Svetltsky and K G Wilson, Phys Rev D 32 (1985) 2736, M Fukuglta and A Ukawa, Phys Rev Lett 55 (1985) 1854, M Fukuglta, Y Oyanagl and A Ukawa, Phys Rev Lett 57(1986)953 [ 161 R E Shrock, Nucl Phys B 267 (1986) 301 [ 171 J Kogut and E Dagatto, preprint ILL-TH-88-9

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